Engineering Tribology Episode 1 Part 6 pptx

Reynolds provided the first analytical proof that a viscousliquid can physically separate two sliding surfaces by hydrodynamic pressure resulting in lowfriction and theoretically zero we

In this chapter the basic principles of hydrodynamic lubrication will be discussed Themechanisms of hydrodynamic film generation and the effects of operating variables such asvelocity, temperature, load, design parameters, etc., on the performance of such films areoutlined This will be explained using bearings commonly found in many engineeringapplications as examples Secondary effects in hydrodynamic lubrication such as viscousheating, compressible and non-Newtonian lubricants, bearing vibration and deformation,will be described and their influence on bearing performance assessed.

4.2 REYNOLDS EQUATION

The serious appreciation of hydrodynamics in lubrication started at the end of the 19thcentury when Beauchamp Tower, an engineer, noticed that the oil in a journal bearingalways leaked out of a hole located beneath the load The leakage of oil was a nuisance so thehole was plugged first with a cork, which still allowed oil to ooze out, and then with a hardwooden bung The hole was originally placed to allow oil to be supplied into the bearing toprovide ‘lubrication’ When the wooden bung was slowly forced out of the hole by the oil,Tower realized that the oil was pressurized by some as yet unknown mechanism Towerthen measured the oil pressure and found that it could separate the sliding surfaces by ahydraulic force [1] At the time of Beauchamp Tower's discovery Osborne Reynolds and othertheoreticians were working on a hydrodynamic theory of lubrication By a most fortunate

coincidence, Tower's detailed data was available to provide experimental confirmation ofhydrodynamic lubrication almost at the exact time when Reynolds needed it The result ofthis was a theory of hydrodynamic lubrication published in the Proceedings of the RoyalSociety by Reynolds in 1886 [2] Reynolds provided the first analytical proof that a viscousliquid can physically separate two sliding surfaces by hydrodynamic pressure resulting in lowfriction and theoretically zero wear.

At the beginning of the 20th century the theory of hydrodynamic lubrication was successfullyapplied to thrust bearings by Michell and Kingsbury and the pivoted pad bearing wasdeveloped as a result The bearing was a major breakthrough in supporting the thrust of aship propeller shaft and the load from a hydroelectric rotor At the present level oftechnology, loads of several thousand tons are carried, at sliding speeds of 10 to 50 [m/s], inhydroelectric power stations The operating surfaces of such bearings are fully separated by alubricating film, so the friction coefficient is maintained at a very low level of about 0.005 andthe failure of such bearings rarely occurs, usually only after faulty operation Reynolds'theory explains the mechanism of lubrication through the generation of a viscous liquid filmbetween the moving surfaces The condition is that the surfaces must move, relatively toeach other, with sufficient velocity to generate such a film It was found by Reynolds andmany later researchers that most of the lubricating effect of oil could be explained in terms ofits relatively high viscosity There are, however, some lubricating functions of an oil asopposed to other liquids which cannot be explained in terms of viscosity and these aredescribed in more detail in Chapter 8 on ‘Boundary and Extreme Pressure Lubrication’.All hydrodynamic lubrication can be expressed mathematically in the form of an equationwhich was originally derived by Reynolds and is commonly known throughout theliterature as the ‘Reynolds equation’ There are several ways of deriving this equation Since

it is a simplification of the Navier-Stokes momentum and continuity equation it can bederived from this basis It is, however, more often derived by considering the equilibrium of

an element of liquid subjected to viscous shear and applying the continuity of flow principle.There are two conditions for the occurrence of hydrodynamic lubrication:

· two surfaces must move relatively to each other with sufficient velocity for a carrying lubricating film to be generated and,

load-· surfaces must be inclined at some angle to each other, i.e if the surfaces are parallel

a pressure field will not form in the lubricating film to support the required load.There are two exceptions to this last rule: hydrodynamic pressure can be generated betweenparallel stepped surfaces or the surfaces can move towards each other (these are special casesand are discussed later) The principle of hydrodynamic pressure generation between movingnon-parallel surfaces is schematically illustrated in Figure 4.1

It can be assumed that the bottom surface, sometimes called the ‘runner’, is covered withlubricant and moves with a certain velocity The top surface is inclined at a certain angle tothe bottom surface As the bottom surface moves it drags the lubricant along it into theconverging wedge A pressure field is generated as otherwise there would be more lubricantentering the wedge than leaving it Thus at the beginning of the wedge the increasingpressure restricts the entry flow and at the exit there is a decrease in pressure boosting the exitflow The pressure gradient therefore causes the fluid velocity profile to bend inwards at theentrance to the wedge and bend outwards at the exit, as shown in Figure 4.1 The generatedpressure separates the two surfaces and is also able to support a certain load It is also possiblefor the wedge to be curved or wrapped around a shaft to form a journal bearing If the wedgeremains planar then a pad bearing is obtained The entire process of hydrodynamic pressuregeneration can be described mathematically to enable accurate prediction of bearingcharacteristics

Pressure profile

p max p

‘dry-water’ The situation dramatically changed with the introduction of computers so thatmechanical systems could be studied in a more detailed fashion

Similarly in hydrodynamics, several simplifying approximations have to be made before amathematical description of the fundamental underlying mechanisms can be derived Allthe simplifying assumptions necessary for the derivation of the Reynolds equation aresummarized in Table 4.1 [3]

The Reynolds equation can now be conveniently derived by considering the equilibrium of

an element (from which the expressions for fluid velocities can be obtained) and continuity

of flow in a column

Equilibrium of an Element

The equilibrium of an element of fluid is considered This approach is frequently used inengineering to derive formulae in stress analysis, fluid mechanics, etc Consider a smallelement of fluid from a hydrodynamic film shown in Figure 4.2 For simplicity, assume that

the forces on the element are acting initially in the ‘x’ direction only.

Since the element is in equilibrium, forces acting to the left must balance the forces acting tothe right, so

τ dxdy x

x

∂τ

∂z ( τ + x dz)dxdy =

pdydz + (p +∂p ∂x dx)dy dz +

(4.1)which after simplifying gives:

∂τ

∂z x dxdydz=

∂p

TABLE 4.1 Summary of simplifying assumptions in hydrodynamics.

Assumption Comments

Always valid, since there are no extra outside fields of forces acting on the fluids with an exception of magnetohydrodynamic fluids and their applications.

Body forces are

neglected

Always valid, since the thickness of hydrodynamic films is

in the range of several micrometers There might be some exceptions, however, with elastic films.

Pressure is constant

through the film

Always valid, since the velocity of the oil layer adjacent to the boundary is the same as that of the boundary.

pdydz

FIGURE 4.2 Equilibrium of an element of fluid from a hydrodynamic film; p is the pressure,

τx is the shear stress acting in the ‘x’ direction.

Assuming that dxdydz ≠ 0 (i.e non zero volume), both sides of equation (4.2) can be divided

by this value and then the equilibrium condition for forces acting in the ‘x’ direction is

A similar exercise can be performed for the forces acting in the ‘y’ (out of the page) direction,

yielding the second equilibrium condition,

In the ‘z’ direction since the pressure is constant through the film (Assumption 2) the

pressure gradient is equal to zero:

expressions are different

Remembering the formula for dynamic viscosity discussed in Chapter 2, the shear stress ‘τ’can be expressed in terms of dynamic viscosity and shear rates:

τx is the shear stress acting in the ‘x’ direction [Pa].

Since ‘u’ is the velocity along the ‘x’ axis, the shear stress ‘τ’ is also acting along this direction Along the ‘y’ (out of the page) direction, however, the velocity is different and consequently

the shear stress is different:

τy is the shear stress acting in the ‘y’ direction [Pa];

v is the sliding velocity in the ‘y’ direction [m/s].

Substituting (4.6) into (4.3) and (4.7) into (4.4), the equilibrium conditions for the forces acting

in the ‘x’ and ‘y’ directions are obtained:

The above equations can now be integrated Since the viscosity of the fluid is constant

throughout the film (Assumption 8) and it is not a function of ‘z’ (i.e η ≠ f(z)), the process of

integration is simple For example, separating the variables in (4.8),

In the general case, there are two velocities corresponding to each of the surfaces ‘U 1 ’ and ‘U 2’

By substituting these boundary conditions into (4.10) the constants ‘C 1 ’ and ‘C 2’ are calculated:

of surfaces ‘Couette velocity’

z

x

FIGURE 4.3 Velocity profiles at the entry of the hydrodynamic film

Continuity of Flow in a Column

Consider a column of lubricant as shown in Figure 4.4 The lubricant flows into the column

horizontally at rates of ‘q x ’ and ‘q y’ and out of the column at rates of (q x + ∂q x

∂ x dx) and

(q y + ∂q y

∂ y dy)per unit length and width respectively In the vertical direction the lubricant

flows into the column at the rate of ‘w 0 dxdy’ and out of the column at the rate of ‘w h dxdy’, where ‘w 0 ’ is the velocity at which the bottom of the column moves up and ‘w h’ is thevelocity at which the top of the column moves up

The principle of continuity of flow requires that the influx of a liquid must equal its effluxfrom a control volume under steady conditions If the density of the lubricant is constant(Assumption 7) then the following relation applies:

Since ‘dxdy ≠ 0’ equation (4.14) can be rewritten as:

∂q x

which is the equation of continuity of flow in a column

Flow rates per unit length, ‘q x ’ and ‘q y’, can be found from integrating the lubricant velocityprofile over the film thickness, i.e.:

+ (U 1 − U 2 ) z

2 2h + U 2 z

q x=

3 −

0 h

which after simplifying gives the flow rate in the ‘x’ direction,

Defining U = U 1 + U 2 and V = V 1 + V 2 and assuming that there is no local variation in

surface velocity in the ‘x’ and ‘y’ directions (i.e U ≠ f(x) and V ≠ f(y)) gives:

Simplifications to the Reynolds Equation

It can be seen that the Reynolds equation in its full form is far too complex for practicalengineering applications and some simplifications are required before it can conveniently beused The following simplifications are commonly adopted in most studies:

· Unidirectional Velocity Approximation

It is always possible to choose axes in such a way that one of the velocities is equal to zero, i.e

V = 0 There are very few engineering systems, in which, for example, a journal bearing

slides along a rotating shaft

· Steady Film Thickness Approximation

It is also possible to assume that there is no vertical flow across the film, i.e w h - w 0 = 0 This

assumption requires that the distance between the two surfaces remains constant during the

operation Some inaccuracy may result from this analytical simplification since mostbearings usually vibrate and consequently the distance between the operating surfacescyclically varies Movement of surfaces normal to the sliding velocity is known as a squeezefilm effect Furthermore, in the case of porous bearings there is always some vertical flow ofoil.

Assuming, however, that there is no vertical flow and w h - w 0 = 0, equation (4.22) can be

written in the form:

‘isoviscous’ model where the thermal effects in hydrodynamic films are neglected Thermal

modification of lubricant viscosity does, however, occur in hydrodynamic films and must beconsidered in a more elaborate and accurate analysis which will be discussed later Assumingthat η = constant equation (4.23) can further be simplified:

· Infinitely Long Bearing Approximation

The simplified Reynolds equation (4.24) is two-dimensional and numerical methods areneeded to obtain a solution Thus, for a simple engineering analysis further simplifyingassumptions are made

It is assumed that the pressure gradient acting along the ‘y’ axis can be neglected, i.e ∂p/∂y = 0 and h ≠ f(y) It is therefore necessary to specify that the bearing is infinitely long in the

‘y’ - direction This approximation is known in the literature as the ‘infinitely long bearing’

or simply ‘long bearing approximation’, and is schematically illustrated in Figure 4.5 It can be said that the pressure gradient acting along the ‘y’ - axis is negligibly small compared to the pressure gradient acting along the ‘x’ - axis This assumption reduces the Reynolds equation

to a one-dimensional form which is very convenient for quick engineering analysis

Since ∂p/∂y = 0, the second term of the Reynolds equation (4.24) is also zero and equation

FIGURE 4.5 Pressure distribution in the long bearing approximation.

Now a boundary condition is needed to solve this equation and it is assumed that at somepoint along the film, pressure is at a maximum At this point the pressure gradient is zero,

i.e dp/dx = 0 and the corresponding film thickness is denoted as ‘h’.

h

¯

p max p

which is particularly useful in the analysis of linear pad bearings Note that the velocity ‘U’

in the convention assumed is negative, as shown in Figure 4.1

· Narrow Bearing Approximation

Finally it is assumed that the pressure gradient acting along the ‘x’ axis is very much smaller than along the ‘y’ axis, i.e.: ∂p/∂x « ∂p/∂y as shown in Figure 4.6 This is known in the literature as a ‘narrow bearing approximation’ or ‘Ocvirk's approximation’ [3] Actually this

particular approach was introduced for the first time by an Australian, A.G.M Michell, in

1905 It was applied to the approximate analysis of load capacity in a journal bearing [10] Asimilar method was also presented by Cardullo [62] Michell observed that the flow in abearing of finite length was influenced more by pressure gradients perpendicular to the sides

of the bearing than pressure gradients parallel to the direction of sliding A formula for thehydrodynamic pressure field was derived based on the assumption that ∂p/∂x « ∂p/∂y Thiswork was severely criticized by other workers for neglecting the effect of pressure variation in

the ‘x‘ direction when equating for flow in the axial or ‘y’ direction and the work was ignored

for about 25 years as a result of this initial unenthusiastic reception Ocvirk and Dubois laterdeveloped the idea extensively in a series of excellent papers and Michell's approximationhas since gained general acceptance

The utility of this approximation became apparent as journal bearings with progressivelyshorter axial lengths were introduced into internal combustion engines Advances in bearingmaterials allowed the reduction in bearing and engine size, and furthermore the reduction

in bearing dimensions contributed to an increase in engine ratings The axial length of thebearing eventually shrank to about half the diameter of the journal and during the 1950'sthis caused a reconsideration of the relative importance of the various terms in the Reynoldsequation During this period, Ocvirk realized the validity of considering the pressuregradient in the circumferential direction to be negligible compared to the pressure gradient inthe axial direction This approach later became known as the Ocvirk or narrow journalbearing approximation An infinitely narrow bearing is schematically illustrated in Figure4.6 The bearing resembles a well deformed narrow pad Also the film geometry is similar tothat of an ‘unwrapped’ film from a journal bearing, which will be discussed later

FIGURE 4.6 Pressure distribution in the narrow bearing approximation

In this approximation since L « B and ∂p/∂x « ∂p/∂y, the first term of the Reynolds equation

(4.24) may be neglected and the equation becomes: